SYNOPSIS

SUBROUTINE CLAIC1(

JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )

    

INTEGER J, JOB

    

REAL SEST, SESTPR

    

COMPLEX C, GAMMA, S

    

COMPLEX W( J ), X( J )

PURPOSE

CLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that

         twonorm(L*x) = sest
Then CLAIC1 computes sestpr, s, c such that
the vector

                [ s*x ]

         xhat = [  c  ]
is an approximate singular vector of

                [ L     0  ]

         Lhat = [ w' gamma ]
in the sense that

         twonorm(Lhat*xhat) = sestpr.
Depending on JOB, an estimate for the largest or smallest singular value is computed.
Note that [s c]' and sestpr**2 is an eigenpair of the system
    diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
                                          [ conjg(gamma) ] where alpha = conjg(x)'*w.

ARGUMENTS

JOB (input) INTEGER

= 1: an estimate for the largest singular value is computed.
= 2: an estimate for the smallest singular value is computed.

J (input) INTEGER

Length of X and W

X (input) COMPLEX array, dimension (J)

The j-vector x.

SEST (input) REAL

Estimated singular value of j by j matrix L

W (input) COMPLEX array, dimension (J)

The j-vector w.

GAMMA (input) COMPLEX

The diagonal element gamma.

SESTPR (output) REAL

Estimated singular value of (j+1) by (j+1) matrix Lhat.

S (output) COMPLEX

Sine needed in forming xhat.

C (output) COMPLEX

Cosine needed in forming xhat.